Probably certifiably correct k-means clustering

Recently, Bandeira (C R Math, 2015 ) introduced a new type of algorithm (the so-called probably certifiably correct algorithm) that combines fast solvers with the optimality certificates provided by convex relaxations. In this paper, we devise such an algorithm for the problem of k -means clustering. First, we prove that Peng and Wei’s semidefinite relaxation of k -means Peng and Wei (SIAM J Optim 18(1):186–205, 2007 ) is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next, we show how to test the optimality of a proposed k -means solution using this dual certificate in quasilinear time. Finally, we analyze a version of spectral clustering from Peng and Wei (SIAM J Optim 18(1):186–205, 2007 ) that is designed to solve k -means in the case of two clusters. In particular, we show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model.